# Practice Test

Q1) A vector which is collinear / coincident / parallel with any given vector is Show Answer

Q2) Two vectors are collinear / coincident / parallel if each one of them is Show Answer

Q3) [ i, j, k] = Show Answer

Q4) If vectors i + j + k, i - j + k and 2i + 3j + mk are coplanar, then m = Show Answer

Q5) If the origin O and the points A(1, 2, 3), B(2, 3, 4) and, P(x,y,z) are coplanar then Show Answer

Q6) i . ( j x k ) + j . ( k x i ) + k . ( i x j ) = Show Answer

Q7) If vectors 2i - j + k, i + 2j - 3k and 3i + mj + 5k are coplanar, then m is a root of the equation Show Answer

Q8) If vectors 2i - j + k, i + 2j - 3k and 3i + aj + 5k are coplanar then a = Show Answer

Q9) Vectors i + j + (m + 1) k, i + j + mk and i - j + mk are coplanar for Show Answer

Q10) Which of the following is not equal to any of the remaining three ? Show Answer

Q11) 2i.(j x k) - 3j.(i x k) - 4k.(i x j) = Show Answer

Q12) If c is the mid-point of AB, and P is a point outside AB, then Show Answer

Q13) If vectors ai + j + k, i - bj + k, i + j - ck are co-planar, then abc + 2 = Show Answer

Q14) If the four points A(2-x, 2, 2), B(2, 2-y, 2), C(2, 2, 2-z) and D(1, 1, 1) are co-planar, then Show Answer

Q15) If the vectors i + j, i - j and li + mj+ nk are coplanar, then: Show Answer

Q16) Points A(4, 5, 1), B(0, -1, 1), C(3, 9, 4) and D(-4, 4, 4) are Show Answer

Q17) A unit vector which is coplanar with (i + j + 2k) and ( i + 2j + k) and perpendicular to (i + j + k) is Show Answer

Q18) If direction of ratios of two lines are 2, -6, -3 and 4, 3, -1 then direction ratios of a line perpendicular to both of them are Show Answer

Q19) If a and b are two collinear vectors, then which of the following are incorrect ? Show Answer

Q20) The points A (1,2,7), B (2,6,3) and C (3,10,-1) are Show Answer

Q21) If a = b + c , then which of the following statements is correct ? Show Answer

Q22) Find the position vector of point R which divides the line joining two points P (2a + b) and Q (a - 3b) externally in the ratio 1 : 2. Also show that P is the middle point of the line segment RQ. Show Answer

Q23) If the points (-1, -1, 2) , (2, m, 5) and (3, 11, 6) are collinear, then the value of m is Show Answer

Q24) If a and b are the position vectors of A and B, respectively, then the position vector of a point C in BA produced such that BC = 1.5 BA, is Show Answer

Q25) If a + b + c = 0, then which of the following is correct ? Show Answer

Q26) The value of [a - b b - c c - a] is equal to Show Answer

Q27) For any three vectors a, b and c the value of [a + b b + c c + a] is equal to Show Answer

Q28) If a and b are two non-collinear vectors and xa + yb = 0 Show Answer

Q29) Five points given by A, B, C, D and E are in a plane. Three forces AC, AD and AE act at A and three forces CB, DB, EB act at B. Then their resultant is Show Answer

Q30) If ABCDEF is regular hexagon, then AB + EB + FC is equal to Show Answer

Q31) If position vector of a point A is
a + 2b and any point P(a) divides AB in the ratio of 2 : 3, then position vector of B is Show Answer

Q32) a, b and c are mutually perpendicular unit vectors, then | a + b + c | is equal to Show Answer

Q33) If A, B, C, D and E are five coplanar points, then DA + DB + DC + AE + BE + CE is equal to Show Answer

Q34) The moment about the point M (-2, 4, -6 ) of the force represented in magnitude and position AB, where the points A and B have the coordinators (1, 2, -3) and ( 3, -4, 2 ) respectively is Show Answer

Q35) If x . a = x . b = x . c = 0, where x is non-zero vector.
Then, [ a x b b x c c x a ] is equal to Show Answer

Q36) a x [ a x ( a x b )] is equal to Show Answer

Q37) [ b x c c x a a x b ] is equal to Show Answer

Q38) If a, b and c are the three vectors mutually perpendicular to each other to form a right handed system and |a| = 1, |b| = 3 and |c| = 5, then [ a - 2b b - 3c c - 4a ] is equal to Show Answer

Q39) If a, b and c are unit coplanar vectors, then [ 2a - b 2b - c 2c - a ] is equal to Show Answer

Q40) For any three vectors a, b and c, (a - b) . (b + c) x ( c + a) is equal to Show Answer

Q41) What is the value of
(d + a) . [ a x { b x ( c x d )}] ? Show Answer

Q42) A tetrahedron has vertical at
O (0, 0), A ( 1, 2, 1), B (2, 1, 3) and
C ( -1, 1, 2). Then, the angle between the faces OAB and ABC will be Show Answer

Q43) The sum of two unit vectors is a unit vector. The magnitude of their difference is Show Answer

Q44) For non-zero vector a and b, if
| a + b | < | a - b |, then a and b are Show Answer

Q45) If ( a x b ) x c = -5a + 4b and a . b = 3, then a x ( b x c ) is equal to Show Answer

Q46) If a and b are two unit vectors such that a + 2b and 5a - 4b are perpendicular to each other, then the angle between a and b is Show Answer

Q47) If a + b + c = 0 and |a| = 3, |b| = 5 and |c| = 7 , then the angle between a and b is Show Answer

Q48) The vector ( a + 3b ) is perpendicular to ( 7a - 5b ) and ( a- 4b ) is perpendicular to ( 7a - 2b ). The angle between a and b is Show Answer

Q49) If a and b are unit vectors, then the greatest value of |a + b| + |a - b| is Show Answer

Q50) Let u, v and w be three vectors such that |u| = 1, |v| = 2, |w| = 3. If the projection of v along u is equal to that of w along u, v and w are perpendicular to each other, then | u - v + w | is equal to Show Answer

Q51) Vectors a and b are such that |a| = 1, |b| = 4 and a . b = 2. If c = 2a x b - 3b, then the angle between b and c is Show Answer

Q52) If a is a unit vector and projection of x along a is 2 and a x r + b = r, then r is equal to

Q53) If a + 2b + 3c = 0, then
a x b + b x c + c x a = ka x b ,
where k is equal to Show Answer

Q54) a and b are two given vectors. With these vectors as adjacent sides, a parallelogram is constructed. The vector which is the altitude of the parallelogram and which is perpendicular to a is Show Answer

Q55) Let a and b be two non - zero perpendicular vectors. A vector r satisfying the equation r x b = a can be Show Answer

Q56) Vector x is Show Answer

Q57) Vector y is Show Answer

Q58) Vector z is Show Answer

Q59) The position vector of P is Show Answer

Q60) The volume of the tetrahedron ABCF is Show Answer

Q61) Let r be a non-zero vector satisfying r . a = r . b = r . c = 0 for given non-zero vectors a, b and c.
Statement I [ a - b b - c c - a ] = 0
Statement II [ abc ] = 0 Show Answer

Q62) The vectors a and b are not perpendicular c and d are two vectors satisfying b x c = b x d and a . d =0.
Then the vectors d is equal to Show Answer

Q63) Let a, b and c be three non-zero vectors which are pairwise non-collinear. If a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is Show Answer

Q64) Which of the following expressions are meaningful? Show Answer

Q65) The volume of the tetrahedron formed by the points (1, 1, 1), (2, 1, 3), (3, 2, 2) and (3, 3, 4) in cubic units is: Show Answer

Q66) The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vectors is Show Answer

Q67) Which one of the following is not correct? Show Answer

Q68) I. Two non-zero. Non-collinear vectors are linearly independent.
II. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true? Show Answer

Q69) The vector equation of the sphere whose center is the point (1, 0, 1) and radius is 4, is Show Answer